Transolver-sigma: Joint Spectral-Physical Subspace Modeling for Neural PDE Solving

arXiv:2609.37279v1 Announce Type: new Abstract: Neural solvers offer efficient surrogates for numerical simulation of partial differential equations (PDEs). For time-dependent problems, strong one-step accuracy does not necessarily translate into reliable autoregressive rollout. We observe that a solver based only on physical-state modeling can achieve lower one-step error, whereas its spectral-only counterpart can become more accurate at later rollout steps. Motivated by this observation, we present Transolver-sigma, a neural PDE solver based on joint spectral--physical subspace modeling. Within each block, adaptive physical-state interactions and spectral transformations are modeled in dedicated latent subspaces, whose responses are recomposed to enable information exchange between the two representations. Within the physical subspace, we introduce Slice-Residual Physics-Attention (SRPA), which preserves an explicit slice-space identity path while retaining learnable cross-slice interaction. In parallel, an axis-factorized Fourier operator captures global spectral structure. Across five well-established PDE benchmarks spanning steady-state prediction and time-dependent dynamics, Transolver-sigma achieves state-of-the-art with a benchmark-averaged relative error reduction of 33.4% over the strongest baseline for each metric, while consistently improving autoregressive rollout over single-operator counterparts. Transolver-sigma further delivers strong gains on coupled multiphysics systems and real-world fluid and combustion measurements from RealPDEBench, demonstrating its effectiveness beyond standard simulation benchmarks.

Source: arXiv cs.AI | 2026-09-30